Williams Peters
03/03/2023 · Primary School
5. Express each of the following in the form \( \frac{x}{y} \), where \( x \) and \( y \) are integers \( \begin{array}{lll}\text { (i) } 0.1666 \ldots & \text { (ii) } 0 . \overline{24} & \text { (iii) } 0.016 \overline{7}\end{array} \) 6. Prove that the following are irrational numbers; \( \begin{array}{lll}\text { (i) } \sqrt{3} & \text { (ii) } 1+\sqrt{2} & \text { (iii) } \sqrt{3}+\sqrt{2}\end{array} \)
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### Problem 5: Expressing Decimals as Fractions
**(i) \( 0.1666\ldots \)**
\( 0.1666\ldots = \frac{1}{6} \)
**(ii) \( 0.\overline{24} \)**
\( 0.\overline{24} = \frac{8}{33} \)
**(iii) \( 0.016\overline{7} \)**
\( 0.016\overline{7} = \frac{1661}{99000} \)
### Problem 6: Proving Irrationality
**(i) \( \sqrt{3} \)**
\( \sqrt{3} \) is irrational.
**(ii) \( 1 + \sqrt{2} \)**
\( 1 + \sqrt{2} \) is irrational.
**(iii) \( \sqrt{3} + \sqrt{2} \)**
\( \sqrt{3} + \sqrt{2} \) is irrational.
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